Determinant Calculator
Calculate the determinant of square matrices instantly. Supports 1x1 to 6x6 matrices with step-by-step solutions.
Determinant Result
Calculation Steps
Understanding Determinants
What is a determinant?
In linear algebra, the determinant is a scalar value that is a function of the entries of a square matrix. It allows characterizing some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible.
How to calculate a 2x2 determinant
For a 2x2 matrix A = [a b; c d], the formula is straightforward:
det(A) = (a × d) - (b × c)
How to calculate a 3x3 determinant
For a 3x3 matrix, you can use the Laplace expansion along the first row (or any row/column), or the Rule of Sarrus. Using expansion:
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Singular Matrices
If the determinant of a matrix is equal to zero (det = 0), the matrix is called a singular matrix. This means the matrix does not have an inverse, and its corresponding system of linear equations does not have a unique solution.
Frequently Asked Questions
A determinant is a specific number calculated from a square matrix. It provides important information about the matrix, such as whether it has an inverse and the scaling factor of the linear transformation it represents.
Enter your matrix values into our calculator and click "Calculate". For manual calculation, the method depends on the size. 2x2 uses cross multiplication (ad-bc), while larger matrices use Laplace expansion or row reduction.
Yes, this tool can easily calculate 3x3 matrices. Simply select "3 x 3" from the Matrix Size dropdown.
Absolutely. The calculator supports square matrices up to 6x6. It uses efficient algorithms (like Gaussian elimination) to compute large determinants accurately.
If a matrix has a determinant of 0, it is called a "singular matrix". This implies the matrix is not invertible, and its rows or columns are linearly dependent.
Yes, you can input decimal fractions (e.g., 2.5, -0.1). Our calculator handles floating-point arithmetic properly.
Yes, negative numbers are fully supported. Just type the minus sign before the number (e.g., -5).
Yes, this Determinant Calculator is completely free to use online, directly in your browser without any required downloads.